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Concept# Residual (numerical analysis)

Summary

Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that
: f(x)=b.
Given an approximation x0 of x, the residual is
: b - f(x_0)
that is, "what is left of the right hand side" after subtracting f(x0)" (thus, the name "residual": what is left, the rest). On the other hand, the error is
: x - x_0
If the exact value of x is not known, the residual can be computed, whereas the error cannot.
Residual of the approximation of a function
Similar terminology is used dealing with differential, integral and functional equations. For the approximation f_\text{a} of the solution f of the equation
: T(f)(x)=g(x) , ,
the residual can either be the function
: ~g(x)~ - ~T(f_\text{a})(x),
or can be said to be the maximum of the norm of this difference
: \max_{x\in \mathcal X} |g(x)-T(f_\text{a})(x)|
over the domain \mathcal X

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Inclinometry with embedded probes is analyzed with a Stokes model of a solid body floating in a fluid with much smaller viscosity for a two-dimensional flow field. The assumption that such a probe behaves like a Lagrangian unit vector is only justified for probes embedded in a Newtonian fluid with lengths at least four times their width. A fluid with Glen-type rheology results in a slightly smaller rotation rate of the probe compared to Newtonian fluids.

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non-singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single-layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any (infinity) domains. We prove that the properties of stability and convergence remain valid. Copyright (C) 2002 John Wiley Sons, Ltd.

2002In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright 2001 John Wiley & Sons, Ltd.

2001